Decidability in continuous-time dynamical
systems
Emmanuel Hainry
LORIA – Nancy Université – UHP
Équipe Carte
NWC 09
Orléans
1/31
Introduction
Problems
Linear Dynamical Systems
Conclusion
Why study dynamical systems
I
Describe physical phenomena
I
Describe biological phenomena
I
Simulate computation models
The natural questions on those systems are motivated by the
applications:
I
Population extinction −→ the system reaches 0.
I
A cloud goes over a region −→ the trajectory intersects a
region.
I
A program loops infinitely −→ the system is ultimately
periodic.
2/31
Introduction
Problems
Linear Dynamical Systems
Conclusion
Discrete time dynamical systems
Definition
A discrete-time dynamical system is given by (X , f ) where X is a
space, and f a function from X to X .
Given a point xn , the successor is xn+1 = f (xn ).
An initial point describes uniquely the whole trajectory.
3/31
Introduction
Problems
Linear Dynamical Systems
Conclusion
Example : second order linear recurrent sequence
X = Z2
f : (x, y ) 7→ (x + y , x)
A trajectory: (1, 1) → (2, 1) → (3, 2) → (5, 3) → (8, 5) → ...
Can be represented by
x
1 1
x
7→
y
1 0
y
4/31
Introduction
Problems
Linear Dynamical Systems
Conclusion
Example: Turing machine
Given a one tape Turing machine with
δ : N × {0, .., 9} → N × {0, .., 9} × {l, s, r }
X = N × N × N (state, left and right parts of tape)
1 3 2 5? 0 1
is coded by a = 1325 et b = 10
f : (n, a, b) 7→ (n0 , a0 , b0 ) with ab coding for the tape,
σ = a mod 10 ; δ(n, σ) = (n0 , σ 0 , τ )
a0 = a − σ + σ 0 ; b0 = b if τ = r
a0 = a/10 ; b0 = 10 × b + σ 0 if τ = d
b0 = b/10 ; a0 = 10 × (a − σ + σ 0 ) + (b mod 10) if τ =
5/31
Introduction
Problems
Linear Dynamical Systems
Conclusion
Continuous time dynamical systems
Definition
A continuous-time dynamical system is defined by (X , f ) where X
is the configuration space (Rn ) and f : X → X .
A trajectory of the system is a solution of the Cauchy problem:
y 0 = f (y )
y (0) = y0
6/31
Introduction
Problems
Linear Dynamical Systems
Conclusion
Example : Piecewise Constant Derivative
x0
7/31
Introduction
Problems
Linear Dynamical Systems
Conclusion
Example: n-body problem
8/31
Introduction
Problems
Linear Dynamical Systems
Conclusion
n-body problem
Proposition
The n-body problem (with Newton’s laws) can be written as a
polynomial dynamical system (with n2 components)
Theorem [Warren D. Smith]
The n-body problem can “solve” the halting problem for Turing
machines in constant time.
I
This is a polynomial dynamical system.
I
The collapsing of the n-body problem is undecidable.
9/31
Introduction
Problems
Linear Dynamical Systems
Conclusion
Example: Lorenz’ attractor
 0 

10(y − x)
x
y  = 28x − y − xz 
z
xy − 83 z
10/31
Introduction
Problems
Linear Dynamical Systems
Conclusion
General Purpose Analog Computer
gpac [Shannon 41] consists in
circuits interconnecting the
following components:
Computing exp with a GPAC
R
f
g
R
a
t0
a+
Rt
t0
f (u)dg(u)
t
1
0
exp
y0 = y
y (0) = 1
Computing cos with a GPAC
g
f
λ
λ
+
f +g
−1
R
t
g
f
×
y1
R
y2
×
y3
f ×g
11/31
Introduction
Problems
Linear Dynamical Systems
Conclusion
Features of the GPAC
Theorem [Graça Costa 03]
A scalar function f : R → R is generated by a GPAC iff it is a
component of the solution of a system
y 0 = p(t, y ),
(1)
where p is a vector of polynomials.
I
gpac is a polynomial dynamical system.
12/31
Introduction
Problems
Linear Dynamical Systems
Conclusion
Continuous time linear dynamical systems
Definition
A continuous-time linear dynamical system is described by a
dimension n, a sqaure matrix A of size n2 with rationnal
coefficients.
X = Rn
f : Y 7→ AY
A trajectory
from Y0 ∈ Qn is a solution of the Cauchy
issued
0
Y = AY
problem:
. Id est Y (t) = exp(tA)Y0 .
Y (0) = Y0
13/31
Introduction
Problems
Linear Dynamical Systems
Conclusion
Reachability
Definition
Given a dynamical system (X , f ), and two points A and B, does
the trajectory issued from A reach B?
I
Safety problem.
14/31
Introduction
Problems
Linear Dynamical Systems
Conclusion
ω-limit set
Definition
Given a dynamical system with solution y , the ω-limit set is the set
of A ∈ X such there (tn ) → +∞ such that lim y (tn ) = A.
I
Périodicity, divergence.
15/31
Introduction
Problems
Linear Dynamical Systems
Conclusion
Skolem-Pisot problem
Definition
Given a recurrent linear sequence, is it sometimes 0?
This problem is equivalent to “given a matrix A ∈ Nn , does the
dynamical system (Nn , Y 7→ AY ) reach a (0, , ..., ) point?”
I
Reachability of a hyperplane
16/31
Introduction
Problems
Linear Dynamical Systems
Conclusion
Undecidability
Theorem
Reachability is undecidable
The halting problem can be written as a reachability question.
17/31
Introduction
Problems
Linear Dynamical Systems
Conclusion
Undecidability
Proposition
Reachability is undecidable in Polynomial DS.
Hyperplane reachability is undecidable in Polynomial DS.
Proof: From [Graça, Campagnolo, Buescu 2005], Turing machines
can be simulated by continuous time polynomial dynamical
systems.
18/31
Introduction
Problems
Linear Dynamical Systems
Conclusion
Simulating Turing machine with a continuous DS
Dynamical System on R3 (state, left and right parts of the tape)
f : N3 → N3 describing the Turing machine.
Duplicate the state space to simulate the transition:
12
∂y1
3
∂t = λ(f (int(y2 )) − y1 ) θ(sin(2πt))
∂y2
3
∂t = λ(int(y1 ) − y2 ) θ(− sin(2πt))
y1 (x, 0) = x
y2 (x, 0) = x
11
10
9
8
7
6
5
4
3
2
with θ Heaviside’s function.
1
0
0,5
1
1,5
2
2,5
3
3,5
4
19/31
Introduction
Problems
Linear Dynamical Systems
Conclusion
Decidability
Proposition [Halava, Harju, Hirvensalo, Karhumäki 2005]
For small dimensions (≤ 5), Skolem-Pisot’s problem is decidable.
Proposition [Blondel, Portier 2003]
Pisot’s problem is NP-hard.
It is unknown whether it is decidable or not for dimension higher
than 5.
20/31
Introduction
Problems
Linear Dynamical Systems
Conclusion
Facts in discrete DS
For discrete time dynamical systems,
ω-limit set
reachability
DS
non computable undecidable
polynomial DS non computable undecidable
linear DS
decidable
hyperplane reach.
undecidable
undecidable
Pisot : open
21/31
Introduction
Problems
Linear Dynamical Systems
Conclusion
Facts in continuous DS
For continuous-time dynamical systems,
ω-limit set
Reachability
DS
non computable undecidable
polynomial DS non computable undecidable
deg.2 poly DS non computable undecidable
linear DS
computable
decidable
hyperplane reach.
undecidable
undecidable
undecidable
?
22/31
Introduction
Problems
Linear Dynamical Systems
Conclusion
Reachability
Theorem [Hai08a]
Reachability is decidable in continuous-time linear dynamical
systems.
f :
Rn → Rn
X 7→ A · X with A ∈ Qn×n
X0 initial point
Y target
23/31
Introduction
Problems
Linear Dynamical Systems
Conclusion
Omega-limit set
Theorem [Hai08b]
The ω-limit set is computable for continuous-time linear dynamical
systems.
Theorem
The ω-limit set for a continuous-time linear dynamical system is
semi-algebraical.
f :
Rn → Rn
X 7→ A · X with A ∈ Qn×n
X0 : initial point
Ω: ω-limit set
24/31
Introduction
Problems
Linear Dynamical Systems
Conclusion
Prerequisite
Theorem [Baker]
Given α ∈ C? , either α or exp(α) is transcendantal.
Theorem [Gelfond-Schneider]
Let α and β algebraic, if α ∈
/ {0, 1} and β ∈
/ Q, then αβ is
transcendantal.
Definition
An algebraic number x is represented by its minimal polynomial χ,
a and such that x is the only root of χ in B(a, )
Proposition
+, −, ×, / are computable for algebraic numbers.
Deciding whether an algebraic number is rational is decidable.
25/31
Introduction
Problems
Linear Dynamical Systems
Conclusion
Simple case
If the matrix A is diagonal, everything is simple : the yi (t) are
independent and yi (t) = exp(tλi )yi0 .
If λi > 0, the ω-limit set is empty.
Otherwise, the ω-limit set is a singleton {yi? } with
yi? = 0
if yi0 = 0
?
yi = 0
if λi < 0
yi? = yi0 otherwise
Reachability is also a disjonction of simple cases.
26/31
Introduction
Problems
Linear Dynamical Systems
Conclusion
General case
In the general case, we transform the matrix to a form close to
diagonal matrix: Jordan form


D1 0
0

.. 
 0 D2 0
. 


A= .

..
 ..

.
0 · · · 0 Dk


λ
1 λ



Di = 

.
.
.
.


.
.
1 λ


B
I2 B

a −b
1


or Di = 
et I2 =
 avec B =
.
.
.
.
b a
0


.
.
I2 B
a
0
1
27/31
Introduction
Problems
Linear Dynamical Systems
Conclusion
General case
Then the difficult cases are those when different components follow
circular trajectories.
Using Baker’s and Gelfond’s theorem, it is possible to rule out
many impossible cases.
In the end, a few algebraic expression describe the ω-limit set.
A few algebraic tests allow to answer the reachability question.
See [Hai08a, Hai08b]
28/31
Introduction
Problems
Linear Dynamical Systems
Conclusion
Continuous Skolem-Pisot
The continuous Skolem-Pisot problem (reachability of a
hyperplane) is still open.
Some cases can be reduced to the existence of a positive root to a
polynomial (see [BDJB08]).
But many cases are not yet shown as decidable or undecidable.
29/31
Introduction
Problems
Linear Dynamical Systems
Conclusion
Conclusion
As for discrete DS, reachability is decidable for continuous-time
linear Dynamical Systems but undecidable for polynomial DS.
Also, the ω-limit set is computable for continuous linear DS.
What about Skolem-Pisot? Reachability of a polyedron?
30/31
O. Bournez, M. L. Campagnolo, D. S. Graça, and E. Hainry.
Polynomial differential equations compute all real computable
functions on computable compact intervals.
Journal of Complexity, 23(3):317–335, 2007.
P. Bell, J.-C. Delvenne, R. Jungers, and V. D. Blondel.
The continuous Skolem-Pisot problem: On the complexity of
reachability for linear ordinary differential equations,
2008, http://arxiv.org/abs/0809.2189.
D. S. Graça, M. L. Campagnolo, and J. Buescu.
Robust simulations of Turing machines with analytic maps and
flows.
In CiE 2005, vol. 3526 of LNCS, 169–179. 2005.
E. Hainry.
Computing omega-limit sets in linear dynamical systems.
In UC 2008, Vienna, vol. 5204 of LNCS, 83–95, 2008.
E. Hainry.
Reachability in linear dynamical systems.
In CiE 2008, Athens, vol. 5028 of LNCS, 241–250, 2008.